library(pracma)
A symmetric matrix is a matrix A such that t(A) = A. Such a matrix is necessarily square. ts main diagonal entries are arbitraty, but its other entries occur in pairs - on opposite sides of the main diagonal.
Example 1
a_symmetric_matrix <- matrix(c(0, -1, 0, -1, 5, 8, 0, 8, -7), nrow = 3, byrow = TRUE)
a_nonsymmetric <- matrix(c(1, -4, 0, -6, 1, -4, 0, -6, 1), nrow = 3, byrow = TRUE)
t(a_symmetric_matrix)==a_symmetric_matrix
## [,1] [,2] [,3]
## [1,] TRUE TRUE TRUE
## [2,] TRUE TRUE TRUE
## [3,] TRUE TRUE TRUE
t(a_nonsymmetric)==a_nonsymmetric
## [,1] [,2] [,3]
## [1,] TRUE FALSE TRUE
## [2,] FALSE TRUE FALSE
## [3,] TRUE FALSE TRUE
Example 2
Review of diagonalization of matrix.
Produce a Basis for each eigenspace.
A <- matrix(c(6, -2, -1, -2, 6, -1, -1, -1, 5), nrow = 3, byrow = TRUE)
A_info <- eigen(A)
# Let P be
P <- A_info$vectors
P
## [,1] [,2] [,3]
## [1,] 7.071068e-01 -0.4082483 -0.5773503
## [2,] -7.071068e-01 -0.4082483 -0.5773503
## [3,] -1.947963e-17 0.8164966 -0.5773503
# Show Eigenvalues
A_info$values
## [1] 8 6 3
# Let D be
D <- matrix(c(8, 0, 0, 0, 6, 0, 0, 0, 3), nrow = 3, byrow = TRUE)
D
## [,1] [,2] [,3]
## [1,] 8 0 0
## [2,] 0 6 0
## [3,] 0 0 3
A
## [,1] [,2] [,3]
## [1,] 6 -2 -1
## [2,] -2 6 -1
## [3,] -1 -1 5
P%*%D%*%solve(P)
## [,1] [,2] [,3]
## [1,] 6 -2 -1
## [2,] -2 6 -1
## [3,] -1 -1 5
P%*%D%*%t(P)
## [,1] [,2] [,3]
## [1,] 6 -2 -1
## [2,] -2 6 -1
## [3,] -1 -1 5
Theorem 1:
If A is symmetric, then any two eigenvectors from different eigenspaces are orthogonal.
Theorem 2:
An nXn matrix A is orthogonally diagonalizable if and only if A is a symmetric matrix.
Example 3:
Orthogonally diagonalize the matrix A, where A is equal to
A <- matrix(c(3, -2, 4, -2, 6, 2, 4, 2, 3), nrow = 3, byrow = TRUE)
A
## [,1] [,2] [,3]
## [1,] 3 -2 4
## [2,] -2 6 2
## [3,] 4 2 3
whose characteristic equation is -(x-7)^2*(x+2)
A_info <- eigen(A)
A_info$values
## [1] 7 7 -2
Then, normalize the vectors.
P <- matrix(c(1/sqrt(2),-1/sqrt(18), -2/3, 0, 4/sqrt(18), -1/3, 1/sqrt(2), 1/sqrt(18), 2/3), nrow = 3, byrow = TRUE)
P
## [,1] [,2] [,3]
## [1,] 0.7071068 -0.2357023 -0.6666667
## [2,] 0.0000000 0.9428090 -0.3333333
## [3,] 0.7071068 0.2357023 0.6666667
D <- matrix(c(7, 0, 0, 0, 7, 0, 0, 0, -2), nrow = 3, byrow = TRUE)
Theorem 3: The Spectral Theorem for Symmetric Matrices
An nXn symmetric matrix A has the following properties:
Here are a few exercices corresponding to the information above.
n
Useful in economics as utility functions and in statistics as confidence ellipsoids.
A quadratic form on Rn is a function Q defined on Rn whose vlue at a vector x in Rn can be computed by and expression of the form Q(x) = t(x)%%I%%x = ||x||^2.. Examples 1 and show the connection between any symmetric matrix A and the quadratic form t(x)%%A%%x.
A <- matrix(c(1,-1,-2,2,2,-2), nrow = 3, byrow = TRUE)
svd(A)
## $d
## [1] 4.242641 0.000000
##
## $u
## [,1] [,2]
## [1,] -0.3333333 0.6666667
## [2,] 0.6666667 0.6666667
## [3,] -0.6666667 0.3333333
##
## $v
## [,1] [,2]
## [1,] -0.7071068 0.7071068
## [2,] 0.7071068 0.7071068